Assume there is \(s_{i}\rightarrow s_{0} \in (-1,1)\)  with
\(\lim _{i\rightarrow \infty } (M_{s_{i}}) > (M_{s_{0}}) - \delta .\) 
Fix \(\mathcal {M}_{i} \in \mathfrak {F}(M_{s_{i}})\)  and \((\mathbf {x}_{i},t_{i})\in \operatorname{sing}\mathcal {M}_{i}\setminus \operatorname{sing}_\textrm {gen}\mathcal {M}_i\)  with
\(\lim _{i\rightarrow \infty }\Theta _{\mathcal {M}_{i}}(\mathbf {x}_{i},t_{i}) > (M_{s_{0}}) - \delta \) 
Pass to a subsequence \(\mathcal {M}_{i}\)  converging to \( \mathcal {M}\in \mathfrak {F}(M_{s_{0}})\)  and \((\mathbf {x}_{i},t_{i})\rightarrow (\mathbf {x}_0,t_0) \in \operatorname{sing}\mathcal {M}\) .
Observe that that if \((\mathbf {x}_0,t_0)\in \operatorname{sing}_\textrm {gen}\mathcal {M}\) , then Proposition REF  implies that
\(\lim _{i\rightarrow \infty } (M_{s_{i}}) \le (M_{s_{0}}) - \delta _0\) 
and we chose \(\delta < \delta _0\)  in Proposition REF . This is a contradiction, so it must hold that \((\mathbf {x}_0,t_0) \in \operatorname{sing}\mathcal {M}\setminus \operatorname{sing}_\textrm {gen}\mathcal {M}\) .
Translate \((\mathbf {x}_0,t_0)\)  to the space-time origin and parabolically dilate to yield \(\tilde{\mathcal {M}}_i\)  and \((\tilde{\mathbf {x}}_i,\tilde{t}_i)\)  with \(|(\tilde{\mathbf {x}}_i,\tilde{t}_i)|=1\)  and
\(\lim _{i\rightarrow \infty }\Theta _{\tilde{\mathcal {M}}_{i}}(\tilde{\mathbf {x}}_{i},\tilde{t}_{i}) > (M_{s_{0}}) - \delta \) 
Pass to a subsequence so that \(\tilde{\mathcal {M}}_i\rightharpoonup \tilde{\mathcal {M}}\)  and \((\tilde{\mathbf {x}}_i,\tilde{t}_i) \rightarrow (\tilde{\mathbf {x}},\tilde{t}) \in (\mathbb {R}^{n+1}\times \mathbb {R})\setminus (\mathbf {0},0)\) . By upper semicontinuity of density
\(\Theta _{\tilde{\mathcal {M}}}(\tilde{\mathbf {x}},\tilde{t}) > (M_{s_{0}}) - \delta .\) 
On the other hand, we can perform the same translation and parabolic dilation to \(\mathcal {M}\)  and by extracting a further subsequence, the resulting flows converge to a tangent flow to \(\mathcal {M}\)  at \((\mathbf {x}_0,t_0)\) . By Lemma REF , the tangent flow is the multiplicity-one flow associated to a smooth shrinker \(\Sigma \) . Note that
\(F(\Sigma ) \le \lambda (\mathcal {M}) \le \limsup _{s\rightarrow s_0} \lambda (M_s) \le \Lambda -\varepsilon .\) 
Since \((\mathbf {x}_0,t_0) \in \operatorname{sing}\mathcal {M}\setminus \operatorname{sing}_\textrm {gen}\mathcal {M}\)  it must hold that \(\Sigma \in \mathcal {S}^*_n(\Lambda -\varepsilon )\setminus \mathcal {S}^\textrm {gen}_n\) . Huisken's monotonicity formula implies that \(\lambda (\tilde{\mathcal {M}}) \le F(\Sigma ) =\Theta _\mathcal {M}(\mathbf {x}_0,t_0)\)  (cf. the proof of Proposition 10.6 in [1]}). Finally, since the supports of \(\mathcal {M}\)  and \(\mathcal {M}_i\)  are disjoint (by the avoidance principle for Brakke flows [2]}), \(\mathcal {M}_i\)  does not smoothly cross \(\mathcal {M}\) . As such (using Lemma REF ), \(\tilde{\mathcal {M}}\)  does not smoothly cross \(t\mapsto ^n\lfloor \sqrt{-t}\Sigma \) . We can now apply Proposition REF  to conclude that
\(\Theta _{\tilde{\mathcal {M}}}(\tilde{\mathbf {x}},\tilde{t}) \le F(\Sigma ) - \delta = \Theta _\mathcal {M}(\mathbf {x}_0,t_0) - \delta \le (M_{s_0}) - \delta .\) 
This contradicts (REF ), completing the proof.
